Polyhedroille

Polyhedroille is a hypothetical class of polyhedral surfaces defined as closed, three-dimensional solids composed of polygonal faces joined edge to edge. Unlike conventional polyhedra, a polyhedroille may admit local variation in face types and vertex arrangements while preserving global connectivity. The term appears in speculative geometry and in educational settings to illustrate how different tiling rules influence a solid's geometry and topology.

Formally, a polyhedroille is specified by a finite set of faces, edges, and vertices that satisfy standard incidence relations: every edge is shared by exactly two faces, and around each vertex follows a prescribed vertex figure (the cyclic sequence of incident face types). The overall topology is captured by Euler’s characteristic χ = V − E + F, which depends on the surface’s genus (for example χ = 2 for a sphere).

Variants range from regular polyhedroilles, with identical face types and vertex figures, to semi-regular families that employ a small set of faces but retain uniform vertex configurations. Higher-genus polyhedroilles admit multiple handles, adjusting χ accordingly.

Applications are primarily pedagogical and exploratory, used to visualize tiling dynamics, explore symmetry, or guide lightweight architectural or ventilation structures in design simulations.

See also: Polyhedron, Tilings, Geodesic dome. Note: polyhedroille is a fictional or speculative construct not part of standard geometric literature.